Arbitrary Degree Distribution Research Articles

Eigenvalue spectra of finely structured random matrices.

Random matrix theory allows for the deduction of stability criteria for complex systems using only a summary knowledge of the statistics of the interactions between components. As such, results like the well-known elliptical law are applicable in a myriad of different contexts. However, it is often assumed that all components of the complex system in question are statistically equivalent, which is unrealistic in many applications. Here we introduce the concept of a finely structured random matrix. These are random matrices with element-specific statistics, which can be used to model systems in which the individual components are statistically distinct. By supposing that the degree of "fine structure" in the matrix is small, we arrive at a succinct "modified" elliptical law. We demonstrate the direct applicability of our results to the niche and cascade models in theoretical ecology, as well as a model of a neural network, and a directed network with arbitrary degree distribution. The simple closed form of our central results allow us to draw broad qualitative conclusions about the effect of fine structure on stability.

Higher-order random network models

Most existing random network models that describe complex systems in nature and society are developed through connections that indicate a binary relationship between two nodes. However, real-world networks are so complicated that we can only identify many critical hidden structural properties through higher-order structures such as network motifs. Here we propose a framework in which we define higher-order stubs, higher-order degrees, and generating functions for developing higher-order complex network models. Then we develop higher-order random networks with arbitrary higher-order degree distributions. The developed higher-order random networks share critical structural properties with real-world networks, but traditional connection-based random networks fail to exhibit these structural properties. For example, as opposed to connection-based random network models, the proposed higher-order random network models can generate networks with power-law higher-order degree distributions, right-skewed degree distributions, and high average clustering coefficients simultaneously. These properties are also observed on the Internet, the Amazon product co-purchasing network, and collaboration networks. Thus, the proposed higher-order random networks are necessary supplements to traditional connection-based random networks.

Acquaintance immunization with limited knowledge of network structure

Optimal and efficient immunization of large networks remains a challenging task. Many theories and approaches have been suggested, however most of them require complete knowledge of the underlying network structure. Here, we study a targeted immunization strategy that incorporates the fact that there is often limited knowledge on the network structure. Previous work has suggested ‘acquaintance’ immunization, where rather than selecting a random individual to immunize, an individual is selected and then one of their acquaintances is immunized. Here, we generalize acquaintance immunization to the case where rather than selecting a random acquaintance, we examine the degrees of n acquaintances and immunize the one with the highest degree. We develop and solve an analytic framework for this model and verify our model with extensive numerical simulations. We determine the critical percolation threshold pc and the size of the giant component, , for arbitrary degree distributions. We also consider our immunization strategy on real-world networks and determine the variation of pc with increasing n. We find that our new approach improves on both acquaintance immunization and random immunization using limited knowledge.

Impact of contact heterogeneity on initial growth behavior of an epidemic: Complex network-based approach

The initial growth behavior of the scalar susceptible-infected-recovered (SIR) epidemic model is fully determined by the basic reproduction number. However, increasing individuals’ contact heterogeneity may invalidate the classical results and cause complex dynamics. Thus, we first consider the SIR model in annealed networks with bimodal degree distribution and derive some sufficient or necessary conditions that determine the monotonicity of densities of infected individuals in each degree class around the initial time t=0. Then, we consider the SIR model in annealed networks with arbitrary degree distribution and analyze the initial growth behavior of it. Interestingly, if we assume that initial densities of infected individuals in each degree class are proportional to the right eigenvector of a specified non-negative and irreducible matrix, then the initial growth behavior of infected individuals in each degree class is completely determined by the basic reproduction number. However, this is not the case for any initial condition, and the initial growth behavior may be very complex. Numerical simulations are performed to verify our analytical results and further investigate the effect of contact heterogeneity on disease behavior.

Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise.

The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems.

Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties

The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states (LDOSs). For random graphs with a negative binomial degree distribution, we show that all eigenvectors are extended and that the spectral density exhibits a logarithmic or a power-law divergence when the variance of the degree distribution is large enough. We elucidate this singular behaviour by showing that the distribution of the LDOSs at the centre of the spectrum displays a power-law tail controlled by the variance of the degree distribution. In the regime of weak degree fluctuations the spectral density has a finite support, which promotes the stability of large complex systems on random graphs.

Generalized Mutual Information-Maximizing Quantized Decoding of LDPC Codes With Layered Scheduling

In this paper, we propose a framework of the mutual information-maximizing (MIM) quantized decoding for low-density parity-check (LDPC) codes by using simple mappings and fixed-point additions. Our decoding method is generic in the sense that it can be applied to LDPC codes with arbitrary degree distributions, and can be implemented based on either the belief propagation (BP) algorithm or the min-sum (MS) algorithm. In particular, we propose the MIM density evolution (MIM-DE) to construct the lookup tables (LUTs) for the node updates. The computational complexity and implementation complexity are discussed and compared to the LUT decoder variants. To accelerate the convergence speed of decoding quasi-cyclic LDPC codes, we consider the layered schedule, and develop the layered MIM-DE to design the LUTs based on MS algorithm, leading to the MIM layered quantized MS (MIM-LQMS) decoder. An optimization method is further introduced to reduce the memory requirement for storing the LUTs. Simulation results show that the MIM quantized decoders outperform the state-of-the-art LUT decoders in the waterfall region with both 3-bit and 4-bit precision. Moreover, the 4-bit MIM-LQMS decoder can approach the error performance of the floating-point layered BP decoder within 0:1 dB.

Analysis of Pure Literal Elimination Rule for Non-uniform Random (MAX) k-SAT Problem with an Arbitrary Degree Distribution

MAX k-SAT is one of the archetypal NP-hard problems. Its variation called random MAX k-SAT problem was introduced in order to understand how hard it is to solve instances of the problem on average. The most common model to sample random instances is the uniform model, which has received a large amount of attention. However, the uniform model often fails to capture important structural properties we observe in the real-world instances. To address these limitations, a more general (in a certain sense) model has been proposed, the configuration model, which is able to produce instances with an arbitrary distribution of variables' degrees, and so can simulate biases in instances appearing in various applications. Our overall goal is to expand the theory built around the uniform model to the more general configuration model for a wide range of degree distributions. This includes locating satisfiability thresholds and analysing the performance of the standard heuristics applied to instances sampled from the configuration model. In this paper we analyse the performance of the pure literal elimination rule. We provide an equation that given an underlying degree distribution gives the number of clauses the pure literal elimination rule satisfies w.h.p. We also show how the distribution of variable degrees changes over time as the algorithm is being executed.

Impact of assortative mixing by mask-wearing on the propagation of epidemics in networks

In this study, we discuss the impacts of assortative mixing by mask-wearing on the effectiveness of mask use in suppressing the propagation of epidemics. We employ the mask model, which is an epidemic model involving mask wearers and non-mask wearers. We derive the occurrence probability and mean size of large outbreaks, epidemic threshold, and average epidemic size for the mask model in an assortatively mixed random network that follows an arbitrary degree distribution. Applying our analysis to the Poisson random networks, we find that the assortative (disassortative) mixing by mask-wearing decreases (increases) the epidemic threshold. Assortative mixing, the tendency for (non-)mask wearers to prefer to connect with (non-)mask wearers, is not effective in containing epidemics in that the transmissibility required for large outbreaks to occur is small. On the other hand, in high-transmissibility cases, mask use is most effective in decreasing the occurrence probability and mean size of large outbreaks, as well as the average epidemic size, when the mixing pattern is strongly assortative. Strongly assortative mixing, resulting in the separation of mask wearers and non-mask wearers, reduces the probability and degree of a large outbreak in high-transmissibility cases, although it allows a large outbreak to occur even in low-transmissibility cases. In scale-free networks, mask use is most effective when the mixing pattern is strongly assortative and when it is maximally disassortative, provided that the mask coverage is not low. Both for the Poisson random and scale-free networks, all analytical treatments are in good agreement with simulation results.

Network resilience of non-hub nodes failure under memory and non-memory based attacks with limited information.

Previous studies on network robustness mainly concentrated on hub node failures with fully known network structure information. However, hub nodes are often well protected and not accessible to damage or malfunction in a real-world networked system. In addition, one can only gain insight into limited network connectivity knowledge due to large-scale properties and dynamic changes of the network itself. In particular, two different aggression patterns are present in a network attack: memory based attack, in which failed nodes are not attacked again, or non-memory based attack; that is, nodes can be repeatedly attacked. Inspired by these motivations, we propose an attack pattern with and without memory based on randomly choosing n non-hub nodes with known connectivity information. We use a network system with the Poisson and power-law degree distribution to study the network robustness after applying two failure strategies of non-hub nodes. Additionally, the critical threshold 1 - p and the size of the giant component S are determined for a network configuration model with an arbitrary degree distribution. The results indicate that the system undergoes a continuous second-order phase transition subject to the above attack strategies. We find that 1 - p gradually tends to be stable after increasing rapidly with n. Moreover, the failure of non-hub nodes with a higher degree is more destructive to the network system and makes it more vulnerable. Furthermore, from comparing the attack strategies with and without memory, the results highlight that the system shows better robustness under a non-memory based attack relative to memory based attacks for n > 1. Attacks with memory can block the system's connectivity more efficiently, which has potential applications in real-world systems. Our model sheds light on network resilience under memory and non-memory based attacks with limited information attacks and provides valuable insights into designing robust real-world systems.

Basic reproduction number for the SIR epidemic in degree correlated networks

The basic reproduction number R0 is an important indicator of the severity for an epidemic outbreak, but it may not be obtained easily for heterogeneous populations especially with biased mixing contacts. Usually, explicit formula for the basic reproduction number in degree correlated networks is difficult to get, thus it is very useful to understand the relationship with their uncorrelated counterparts. It seems that the assortative mixing increases the basic reproduction number of susceptible–infected–recovered (SIR) epidemic, whereas disassortative mixing decreases it, or that assortative mixing enhances the percolation, whereas disassortative mixing weakens it. This result is obtained for degree correlated networks based on a small deviation from the uncorrelated networks, and may not be universal. In this paper, we show rigorously that the result is true for degree correlated networks with arbitrary bimodal degree distribution and for assortative mixing networks with arbitrary degree distribution. However, this may not be always true for general disassortative mixing networks. We also introduce a numerical algorithm to construct an arbitrary joint degree distribution and generate a counterexample of disassortative mixing network that the result fails. Finally, a sufficient condition is given to guarantee that the SIR model in disassortative mixing networks yields smaller R0.

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie–Weiss, the Kuramoto, and the Sherrington–Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington–Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree c scales as a power c ∝ N b (b < 1) of the total number N ≫ 1 of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

Epidemics with asymptomatic transmission: Subcritical phase from recursive contact tracing.

The challenges presented by the COVID-19 epidemic have created a renewed interest in the development of new methods to combat infectious diseases, and it has shown the importance of preparedness for possible future diseases. A prominent property of the SARS-CoV-2 transmission is the significant fraction of asymptomatic transmission. This may influence the effectiveness of the standard contact tracing procedure for quarantining potentially infected individuals. However, the effects of asymptomatic transmission on the epidemic threshold of epidemic spreading on networks have rarely been studied explicitly. Here we study the critical percolation transition for an arbitrary disease with a nonzero asymptomatic rate in a simple epidemic network model in the presence of a recursive contact tracing algorithm for instant quarantining. We find that, above a certain fraction of asymptomatic transmission, standard contact tracing loses its ability to suppress spreading below the epidemic threshold. However, we also find that recursive contact tracing opens a possibility to contain epidemics with a large fraction of asymptomatic or presymptomatic transmission. In particular, we calculate the required fraction of network nodes participating in the contact tracing for networks with arbitrary degree distributions and for varying recursion depths and discuss the influence of recursion depth and asymptomatic rate on the epidemic percolation phase transition. We anticipate recursive contact tracing to provide a basis for digital, app-based contact tracing tools that extend the efficiency of contact tracing to diseases with a large fraction of asymptomatic transmission.

Construction of simplicial complexes with prescribed degree-size sequences.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s≥3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young etal., Phys. Rev. E 96, 032312 (2017)2470-004510.1103/PhysRevE.96.032312], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analyses of higher-order phenomena based on local structures.

Degree dependent transmission rates in epidemic processes

The outcome of SIR epidemics with heterogeneous infective lifetimes, or heterogeneous susceptibilities, can be mapped onto a directed percolation process on the underlying contact network. In this paper we study SIR models where heterogeneity is a result of the degree dependence of disease transmission rates. We develop numerical methods to determine the epidemic threshold, the epidemic probability and epidemic size close to the threshold for configuration model contact networks with arbitrary degree distribution and an arbitrary matrix of transmission rates (dependent on transmitting and receiving node degree). For the special case of separable transmission rates we obtain analytical expressions for these quantities. We propose a categorization of spreading processes based on the ratio of the probability of an epidemic and the expected size of an epidemic, and demonstrate that this ratio has a complex dependence on the degree distribution and the degree-dependent transmission rates. For scale-free contact networks and transmission rates that are power functions of transmitting and receiving node degrees, the epidemic threshold may be finite even when the degree distribution powerlaw exponent is below γ < 3. We give an expression, in terms of the degree distribution and transmission rate exponents, for the limit at which the epidemic threshold vanishes. We find that the expected epidemic size and the probability of an epidemic may grow nonlinearly above the epidemic threshold, with exponents that depend not only on the degree distribution powerlaw exponent, but on the parameters of the transmission rate degree dependence functions, in contrast to ordinary directed percolation and previously studied variations of the SIR model.

Efficient Laplacian spectral density computations for networks with arbitrary degree distributions

AbstractThe network Laplacian spectral density calculation is critical in many fields, including physics, chemistry, statistics, and mathematics. It is highly computationally intensive, limiting the analysis to small networks. Therefore, we present two efficient alternatives: one based on the network’s edges and another on the degrees. The former gives the exact spectral density of locally tree-like networks but requires iterative edge-based message-passing equations. In contrast, the latter obtains an approximation of the spectral density using only the degree distribution. The computational complexities are 𝒪(|E|log(n)) and 𝒪(n), respectively, in contrast to 𝒪(n3) of the diagonalization method, where n is the number of vertices and |E| is the number of edges.

Transitions in random graphs of fixed degrees with many short cycles

We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of three-cycles (triangles). We find that such ensembles generally exhibit two transitions, a clustering and a shattering transition, separating three distinct regimes. At the clustering transition, the graphs change from typically having only isolated cycles to forming cycle clusters. At the shattering transition the graphs break up into many small cliques to achieve the desired three-cycle density. The locations of both transitions depend nontrivially on the system size. We derive a general formula for the three-cycle density in the regime of isolated cycles, for graphs with degree distributions that have finite first and second moments. For bounded degree distributions we present further analytical results on cycle densities and phase transition locations, which, while non-rigorous, are all validated via MCMC sampling simulations. We show that the shattering transition is of an entropic nature, occurring for all three-cycle density values, provided the system is large enough.

Ck and C0hp-finite elements on d-dimensional meshes with arbitrary hanging nodes

In this paper, the construction of Ck basis functions is proposed for paraxial d-dimensional rectangular meshes with arbitrary hanging nodes and arbitrary polynomial degree distributions. The construction is based on the large-support approach introduced in [1] for C0 basis functions in 2D and uses hierarchical tensor-product shape functions which combine Hermite shape functions with Gegenbauer polynomials enabling the support of the basis functions to be independent of k (in contrast to basis functions based on B-spline approaches). Moreover, these shape functions allow for an efficient recursive computation of the constraints coefficients in the application of constrained approximation for hanging nodes without the need for collocation. An appropriate indexing of the shape functions is introduced in order to prove the differentiability properties of the basis functions. The construction is also suitable for the extension to C0 finite elements on meshes which are not necessarily rectangular. In particular, the orientation problem resulting from differently oriented edges or faces can be appropriately treated within this extension. Numerical examples illustrate the feasibility of the proposed approach. Moreover, some aspects concerning the condition number of the system matrix resulting from the discretization of Poisson's problem are discussed.

Arbitrary degree distribution networks with perturbations

Complex networks have played an important role in the field of natural science and social science, attracting considerable attention of more and more scholars. Currently, scholars have proposed numbers of complex networks, in which some show a required degree distribution and others follow arbitrary degree distribution. The goal of this paper is to discuss the impact of perturbations on degree distribution. To this end, we first introduce two types of perturbations, i.e., edge perturbations and vertex perturbations, and investigate networks whose structure can be determined by tuning perturbation rules. Next, we calculate the degree distribution using two popularly utilized mathematical methods, namely, rate equation and generating function. Afterward, we analyze several networks with different degree distributions, for example, Poisson distribution, stretched exponential distribution, and power-law distribution; there are, in practice, some pronounced differences among three cases. Therefore, to a certain extent, the above three cases can serve as the measures for degree distribution to help us clearly distinguish among different degree distributions.

Efficient network immunization under limited knowledge.

Targeted immunization of centralized nodes in large-scale networks has attracted significant attention. However, in real-world scenarios, knowledge and observations of the network may be limited, thereby precluding a full assessment of the optimal nodes to immunize (or quarantine) in order to avoid epidemic spreading such as that of the current coronavirus disease (COVID-19) epidemic. Here, we study a novel immunization strategy where only n nodes are observed at a time and the most central among these n nodes is immunized. This process can globally immunize a network. We find that even for small n (≈10) there is significant improvement in the immunization (quarantine), which is very close to the levels of immunization with full knowledge. We develop an analytical framework for our method and determine the critical percolation threshold pc and the size of the giant component P∞ for networks with arbitrary degree distributions P(k). In the limit of n → ∞ we recover prior work on targeted immunization, whereas for n = 1 we recover the known case of random immunization. Between these two extremes, we observe that, as n increases, pc increases quickly towards its optimal value under targeted immunization with complete information. In particular, we find a new general scaling relationship between |pc(∞) − pc(n)| and n as |pc(∞) − pc(n)| ∼ n−1exp(−αn). For scale-free (SF) networks, where P(k) ∼ k−γ, 2 < γ < 3, we find that pc has a transition from zero to nonzero when n increases from n = 1 to O(log N) (where N is the size of the network). Thus, for SF networks, having knowledge of ≈log N nodes and immunizing the most optimal among them can dramatically reduce epidemic spreading. We also demonstrate our limited knowledge immunization strategy on several real-world networks and confirm that in these real networks, pc increases significantly even for small n.